Exploring eigenvectors and eigenvalues in the plane graphicallybob jantzen, villanova university [April, 2023] Vector calculations are normally done using their components, but the geometric information about a vector 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 consists of two separate characteristics: its length |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| and its direction (the latter represented by a unit vector aligned with the original vector: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkmbW92ZXJHRiQ2JS1JI21pR0YkNihRInhGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSSNtb0dGJDYvUScmY2lyYztGJ0Y3LyUscGxhY2Vob2xkZXJHRjQvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZKLyUpc3RyZXRjaHlHRkovJSpzeW1tZXRyaWNHRkovJShsYXJnZW9wR0ZKLyUubW92YWJsZWxpbWl0c0dGSi8lJ2FjY2VudEdGSi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlkvRlZGNC1GQTYuUSJ+RidGN0ZGL0ZJRjkvRkxGOS9GTkY5L0ZQRjkvRlJGOS9GVEY5L0ZWRjlGV0ZaLUZBNi5RIj1GJ0Y3RkZGam5GW29GXG9GXW9GXm9GX29GYG8vRlhRLDAuMjc3Nzc3OGVtRicvRmVuRmVvRmduRjdGRg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUkmbWZyYWNHRiQ2KC1GIzYlLUkmbW92ZXJHRiQ2JS1GIzYpRistRiw2KFEieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGQC8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJ0Y+RkFGQ0ZGRkktSSNtb0dGJDYvUS0mUmlnaHRBcnJvdztGJ0ZDLyUscGxhY2Vob2xkZXJHRkAvRkdRJ25vcm1hbEYnLyUmZmVuY2VHUSZ1bnNldEYnLyUqc2VwYXJhdG9yR0ZWLyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRlYvJShsYXJnZW9wR0ZWLyUubW92YWJsZWxpbWl0c0dGVi8lJ2FjY2VudEdGVi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRl9vL0Zcb0ZARkNGUi1GIzYnLUkobWZlbmNlZEdGJDYnLUYjNictRjc2JUY7RkxGYm9GQUZDRlAvRkdRJ2l0YWxpY0YnRkNGUi8lJW9wZW5HUSJ8Z3JGJy8lJmNsb3NlR0ZgcEZBRkNGUEZccC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGaHAvJSliZXZlbGxlZEdGRUZDRlJGK0ZDRlI=JSFH ) . Matrix multiplication of an input vector 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 by a matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn to obtain an output vector 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changes that geometric information, and the geometric interpretation of those changes can be best understood through those vectors whose direction is not changed by this process, modulo sign, so that the input and output vectors are aligned. These vectors are called eigenvectors of the matrix. The resulting change in length along those special directions are captured in the multipliers, which are the called the eigenvalues corresponding to each eigenvector. The multipliers can be positive or negative depending on whether the output vector points in the same or opposite direction from the input vector, or zero if the output vector is the zero vector.Visualizing the input and output vectors in the plane for matrix multiplication of the position vector is extremely helpful in understanding the meaning of an eigenvector and its eigenvalue. The Maple Explore command allows one to use your mouse to left click and drag the tip of the (red) input vector around on grid points to read off its coordinates as well as those of the (blue) output vector, with the eigenvalue scale factor as a byproduct read off by counting squares of the gridline grid for simple integer cases, or graphically estimated for noninteger eigenvalues. Once an eigenvector representative is found, one can scale it by a suitable positive scale factor to simplify its components, simply by moving the tip of the input vector parallel to itself. For example LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYnLUYjNictSSNtbkdGJDYlUSIyRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiNEYnRjRGN0Y0RjdGNEY3LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0Y0Rjc= can be simplified to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYnLUYjNictSSNtbkdGJDYlUSIxRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiMkYnRjRGN0Y0RjdGNEY3LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0Y0Rjc= corresponding to the line through the origin with slope 2. For noninteger examples, one can estimate approximately the components.This procedure works with single digit matrix entries typical of linear algebra textbook problems. It has taken a few versions of Maple for the mouse dragging feature to work without bugs. This Maple procedure was inspired by the Duke University project Java based interactive eigenvector applet that no longer functions since Big Tech abandoned Java years ago.

This procedure was inspired by the Help page for Explore using the "markers" option. 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

Let's try it with a simple matrix. Execute the command and left click on the input vector tip red circle and mouse drag it around the origin until the lines line up, then scale up or down to use the grid to read off integer components if possible, and the multiplication factor between blue output and red intput vector lengths. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= For example, for this matrix when the red circle is on the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GLDYmUSJ4RidGL0YyRjUtRjk2LlEifkYnRjJGPEY+RkBGQkZERkZGSEZKL0ZNUSYwLjBlbUYnL0ZQRlhGMkY8, they line up, and moving the red vector tip to the point (1,1), the blue circle is then at (4,4), so LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYnLUYjNictSSNtbkdGJDYlUSIxRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY0RjdGNEY3LyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0Y0Rjc= is the simplest integer valued eigenvector, and its eigenvalue is 4. To try it with another matrix, simply edit the input matrix and re-execute. Have fun! Try this is the matrix: 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

6 useful examples of what kinds of situations can arise are found illustrated by direction fields in this worksheet desysplot.mw which shows the 2d plots of the corresponding linear systems of DEs associated with these matrices and the eigenvalues and eigenvectors explicitly. 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 lecture example: 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lecture example: 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